Suppose that
\( \limsup _{n\rightarrow \infty } \mathcal {F}_n(\mathsf {P}^n)=:C < \infty ,\) 
and let us consider the following interaction energy functional:
\(E_g^N(x_1,\dots ,x_N):=\frac{1}{N^2} \sum _{i,j\ne i} |g|(x_i,x_j).\) 
From a Hoeffding's decomposition argument, see [1]}, we have for every \(N\ge 2\) , \(\alpha \ge 0\)  the estimate
\(\frac{1}{N} \log \frac{1}{\gamma (\mathcal {T})^N}\int _{\mathcal {T}^N} e^{\alpha N E_g^N(x_1,\dots ,x_N)} \, \mathrm {d}\gamma ^{\otimes N}\le \genfrac{}{}{}2{1}{2} \log \left(\frac{1}{\gamma (\mathcal {T})^2}\int _{\mathcal {T}\times \mathcal {T}} e^{\frac{2\alpha N}{N-1} |g|(x,y) } \,\mathrm {d}\gamma ^{\otimes 2}\right).\) 

Moreover, since \(N/(N-1)\le 2\)  for \(N\ge 2\) , and
\( \sum _{i,j} |g|(x_i,x_j) = \sum _{i,j\ne i} |g|(x_i,x_j) + \sum _{i} |g|(x_i,x_i) \le \sum _{i,j\ne i} |g|(x_i,x_j) + N \Vert g\Vert _{\infty },\) 
we find that
\(\frac{1}{N} \log \left( \frac{1}{\gamma (\mathcal {T})^N}\int _{\mathcal {T}^N} e^{\frac{\alpha }{N} \sum _{i,j} |g|(x_i,x_j) } \, \mathrm {d}\gamma ^{\otimes N}\right) \le \frac{1}{2}\log \left(\frac{1}{\gamma (\mathcal {T})^2}\int _{\mathcal {T}\times \mathcal {T}} e^{4\alpha |g|(x,y) } \,\mathrm {d}\gamma ^{\otimes 2}\right)+\frac{\alpha \Vert g\Vert _{\infty }}{N}.\) 
Recall that \(L_n(x_1,\dots ,x_N):=\genfrac{}{}{}2{1}{n}\sum \delta _{x_i}\) . Since the mass \(L_n(x_1,\dots ,x_N)(\mathcal {T})=N/n\)  is bounded by \(K\)  on the support of \(F\)  we have for \(N\ge 2\) :
\(|F|(L_n)\le |h|(L_n(\mathcal {T})) \genfrac{}{}{}2{1}{n^2} \sum _{i,j} |g|(x_i,x_j) \le \frac{K \Vert h\Vert _{\infty }}{n N} \sum _{i,j} |g|(x_i,x_j), \) 
while for \(N=1\)  we have the trivial estimate \(|F|(L_n)\le \genfrac{}{}{}2{K}{n} \Vert h\Vert _{\infty } \Vert g\Vert _{\infty }\) , and hence for all \(N\ge 1\) ,
\(\frac{1}{\gamma (\mathcal {T})^N} \int _{\mathcal {T}^N} e^{n |F|(L_N)} \,\mathrm {d}\gamma ^{\otimes N} \le e^{K \Vert g\Vert _{\infty }\Vert h\Vert _{\infty }} \left(\frac{1}{\gamma (\mathcal {T})^2}\int _{\mathcal {T}^2} e^{4K \Vert h\Vert _{\infty } |g|(x,y) } \mathrm {d}\gamma ^{\otimes 2} \right)^{N/2}.\) 
Using the representation for \(\Pi _n\)  we can therefore estimate
\(\int _{\Gamma } e^{n |F|}\, \mathrm {d}\Pi _n&=\frac{1}{e^{n \gamma (\mathcal {T})}-1} \sum _{i=1}^N \frac{(n \gamma (\mathcal {T}))^N}{N!}\int _{\mathcal {T}^N} e^{n |F|} \mathrm {d}\gamma ^{\otimes N}/\gamma (\mathcal {T})^N\\&\le \frac{1}{e^{n \gamma (\mathcal {T})}-1} \sum _{i=0}^N \frac{(n \gamma (\mathcal {T}))^N}{N!} e^{K \Vert g\Vert _{\infty }\Vert h\Vert _{\infty }}\left(\frac{1}{\gamma (\mathcal {T})^2}\int _{\mathcal {T}^2} e^{4K \Vert h\Vert _{\infty } |g|(x,y) } \mathrm {d}\gamma ^{\otimes 2} \right)^{N/2}\\&=\frac{ e^{K \Vert g\Vert _{\infty }\Vert h\Vert _{\infty }}}{e^{n \gamma (\mathcal {T})}-1}\exp \left\lbrace  n\gamma (\mathcal {T}) \left(\frac{1}{\gamma (\mathcal {T})^2}\int _{\mathcal {T}^2} e^{4K \Vert h\Vert _{\infty } |g|(x,y) } \mathrm {d}\gamma ^{\otimes 2} \right)^{1/2} \right\rbrace ,\) 

which proves (REF ). The final desired statement follows directly after taking limits.
